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From Wikipedia

Law of sines

In trigonometry, the law of sines (also known as the sine law, sine formula, or sine rule) is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles. According to the law,

\frac{a}{\sin A} \,=\, \frac{b}{\sin B} \,=\, \frac{c}{\sin C},

where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the opposite angles (see the figure to the right). Sometimes the law is stated using the reciprocal of this equation:

\frac{\sin A}{a} \,=\, \frac{\sin B}{b} \,=\, \frac{\sin C}{c}.

The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known&mdash;a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the formula gives two possible values for the enclosed angle, leading to an ambiguous case.

The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in a general triangle, the other being the law of cosines.

Examples

The following are examples of how to solve a problem using the law of sines:

Given: side a&nbsp;=&nbsp;20, side c&nbsp;=&nbsp;24, and angle C&nbsp;=&nbsp;40Â°

Numeric problems

Like the law of cosines, although the law of sines is mathematically true, it has problems for numeric use. Much precision may be lost if an arcsine is computed when the sine of an angle is close to one.

Some applications

The sine law can be used to prove the angle sum identity for sine when Î± and Î² are each between 0 and 90 degrees.

To prove this, make an arbitrary triangle with sides a, b, and c with corresponding arbitrary angles A, B and&nbsp;C. Draw a perpendicular to c from angle&nbsp;C. This will split the angle C into two different angles, Î± and Î², that are less than 90 degrees, where we choose to have Î± to be on the same side as A and Î² be on the same side as B. Use the sine law identity that relates side c and side&nbsp;a. Solve this equation for the sine of C. Notice that the perpendicular makes two right angles triangles, also note that sin(A) =&nbsp;cos(Î±), sin(B) =&nbsp;cos(Î²) and that c&nbsp;=&nbsp;a&nbsp;sin(Î²)&nbsp;+&nbsp;b&nbsp;sin(Î±). After making these substitutions you should have sin(C) =sin(Î±&nbsp;+&nbsp;Î²) =&nbsp;sin(Î²)cos(Î±)&nbsp;+&nbsp;(b/a)sin(Î±)cos(Î±). Now apply the sine law identity that relates sides b and a and make the substitutions noted before. Now substitute this expression for (b/a) into the original equation for sin(Î±&nbsp;+&nbsp;Î²) and you will have the angle sum identity for Î± and Î² in terms of sine.

The only thing that was used in the proof that was not a definition was the sine law. Thus the sine law is equivalent to the angle sum identity when the angles sum is between 0 and 180 degrees and when each individual angle is between 0 and 90 degrees.

The ambiguous case

When using the law of sines to solve triangles, there exists an ambiguous case where two separate triangles can be constructed (i.e., there are two different possible solutions to the triangle).

Given a general triangle ABC, the following conditions would need to be fulfilled for the case to be ambiguous:

The only information known about the triangle is the angle A and the sides a and b, where the angle A is not the included angle of the two sides (in the above image, the angle C is the included angle).

Spherical case

Here, Î±, Î², and Î³ are the angles at the center of the sphere subtended by the three arcs of the spherical surface triangle a,b, and c, respectively. A, B, and C are the surface angles opposite their respective arcs.